In paper Approximation Algorithms for the Chromatic Sum, page 18, authors state that based on the fact that the Graph Coloring problem is hard to approximate with a ratio less than 2 (under the assumption of $P \neq NP$), it's easy to show that for any constant $k$, there is no approximate algorithm $A$ to Graph Coloring such that $\chi(G) \leq A(G) \leq \chi(G) + k$.
But i can't see how to show/prove this.
As i see, it isn't so direct, because for any $k$, there are graphs such that $\chi(G) \leq k$, so for them $\chi(G) + k \geq 2 \cdot \chi(G)$ and it would be $(2 + \epsilon)$-approximative, for some $\epsilon \geq 0$, for these graphs.
So i'm asking for some help/advice on how to show/prove this.